On the computation of the projective surgery obstruction groups

Abstract

Abstract. The computation of the projective surgery obstruction groups LP. (ZG), for G a hyperelementary finite group, is reduced to standard calculations in number theory, mostly involving class groups. Both the exponent of the torsion subgroup and the precise divisibility of the signatures are determined. For G a 2-hyperelementary group, the LP.(ZG) are detected by restriction to certain subquotients of G, and a complete set of invariants is given for oriented surgery obstructions.Key words. Surgery on manifolds, Hermitian K-theory. O. IntroductionThe projective surgery obstruction groups were first introduced by S. P. Novikov 1-30] in the context of Hermitian K-theory and the topology of infinite cycle covers of compact manifolds. These groups arise as the codimension 1 summands in a splitting theorem for the Wall surgery obstruction groups of a Laurent polynomial extension 1,39, 12], or more generally in the classification of noncompact manifolds [40,41,27,33]. The algebraic description of projective L-theory and the splitting theorem were given a systematic exposition by A. A. Ranicki 1-34, 36], including a definition of the lower L-groups by analogy with the lower K-groups of Bass.With the extensive development of 'bounded' or 'controlled' topology in the last decade, the role of projective and lower surgery obstruction groups has increased in importance. For example, the concrete structure of these groups is relevant to the classification of linear representations of finite groups up to topological conjugacy ~5, 6, 18] and the recent survey article I13] describes other applications. The purpose of the present paper is to provide a reference for further computations in this area.

Splitting along Submanifolds and L-Spectra

“…Approaches to these problems are quite distinct. At the present time, there exist effective methods for calculation of L-groups (see [4][5][6][7][8]) based on results in number theory and algebraic K-theory. Methods of algebraic and geometric topology are used for describing normal invariants and assembly mapping (see [1,3,[9][10][11][12][13]).…”

“…The algebraic construction of diagram (10) for quadratic extensions of anti-structures was given in [33]. In this case, diagram (10) can be efficiently used for calculation of obstruction groups and natural mappings in L-theory (see [7,8,14,34]). Moreover, diagram (10) allows one to obtain deep geometric results on representability of elements of Wall groups by normal mappings of closed manifolds (see [16,17,30,35]).…”

“…Let Ψ be the quadrate of fundamental groups (8) in the problem of splitting along a submanifold Y of codimension q = 1, 2. We denote by B − the oriented left group B.…”

Projective splitting obstruction groups for one-sided submanifolds

“…At present there exist effective methods for the computation of L-groups and LNgroups in the case of finite hyperelementary groups (see, for example, [6], [12]- [15]) and, in particular, in the case of finite 2-groups. These groups occupy a special place in surgery theory since the range of the map σ :…”

“…The most complete results in computation were obtained for the projective L-groups (see [12] and [15]) introduced by Novikov [17].…”